On the support of the free Lie algebra: the Schützenberger problems

We characterize the support of the free Lie algebra LZm(A) on a finite alphabet A over the ring Zm of integers mod m as the set of all words w such that m c(w), where c(w) is the greatest common divisor of the coefficients in the polynomial l∗(w), for the adjoint endomorphism l∗ of the left-normed Lie bracketing l of the free Lie ring. For words of length n this can be interpreted combinatorially as the problem of finding all λ-tabloids t such that ln · t = 0, where ln is the left-normed multilinear Lie bracketing of LZm(A), viewed as an element of the group ring ZmSn of the symmetric group on n letters. For a tabloid with two parts, represented by a subset of I of [n], this boils down to solving pn(I) ≡ 0 ( mod m), for a particular commutative multilinear polynomial pn(I), called Pascal descent polynomial, which for |I| = 1 is just a signed binomial coefficient. An explicit sufficient condition for w to lie in the support of LZm(A) then is m Nn(I), where Nn(I) = X i∈I (−1)i−1 n− 1 i− 1 !